Question

draw a triangle with sides 7cm 6cm 5cm. Then draw of a square of the same area​

Answer 1

Step 1. Construct a rectangle of the same area as the triangle. Euclid's Elements, Book I, Proposition 42. (The area of a triangle is the same as the rectangle of the same height on half its base.) Choose one side of the triangle to call the base of the triangle. Draw a line parallel to that base through the opposite vertex. Draw lines perpendicular to the base at one end of it and at the midpoint of the base. The rectangle you form will have the same area as the triangle.

Step 2. Construct a square equal to that rectangle. Euclid's Elements, Book II, Proposition 14. If the rectangle is ABCD, extend AB by BE equal to BC.

Draw a semicircle on the diameter AE. Let F be the intersection of perpendicular to the diameter at B. Then the square with side BF has the same area as the rectangle, and that has the same area as the original triangle.